Geometry (from the grc|γεωμετρία; ''geo-
'' "earth", ''-metron
'' "measurement") is, with arithmetic
, one of the oldest branches of mathematics
. It is concerned with properties of space that are related with distance, shape, size, and relative position of figures.
A mathematician who works in the field of geometry is called a geometer
Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry
, which includes the notions of point
, and curve
, as fundamental concepts.
During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Gauss
' Theorema Egregium
(remarkable theorem) that asserts roughly that the Gaussian curvature
of a surface is independent from any specific embedding
in an Euclidean space
. This implies that surfaces can be studied ''intrinsically'', that is as stand alone spaces, and has been expanded into the theory of manifold
s and Riemannian geometry
Later in the 19th century, it appeared that geometries without the parallel postulate
) can be developed without introducing any contradiction. The geometry that underlies general relativity
is a famous application of non-Euclidean geometry.
Since then, the scope of geometry has been greatly expanded, and the field has been split in many subfields that depend on the underlying methods—differential geometry
, algebraic geometry
, computational geometry
, algebraic topology
, discrete geometry
(also known as ''combinatorial geometry''), etc.—or on the properties of Euclidean spaces that are disregarded—projective geometry
that consider only alignment of points but not distance and parallelism, affine geometry
that omits the concept of angle and distance, finite geometry
that omits continuity
Often developed with the aim to model the physical world, geometry has applications to almost all science
s, and also to art
, and other activities that are related to graphics
Geometry has also applications to areas of mathematics that are apparently unrelated. For example, methods of algebraic geometry are fundamental for Wiles's proof
of Fermat's Last Theorem
, a problem that was stated in terms of elementary arithmetic
, and remained unsolved for several centuries.
The earliest recorded beginnings of geometry can be traced to ancient Mesopotamia
in the 2nd millennium BC. Early geometry was a collection of empirically discovered principles concerning lengths, angles, areas, and volumes, which were developed to meet some practical need in surveying
, and various crafts. The earliest known texts on geometry are the Egyptian
'' (2000–1800 BC) and ''Moscow Papyrus
'' (c. 1890 BC), the Babylonian clay tablets
such as Plimpton 322
(1900 BC). For example, the Moscow Papyrus gives a formula for calculating the volume of a truncated pyramid, or frustum
Later clay tablets (350–50 BC) demonstrate that Babylonian astronomers implemented trapezoid
procedures for computing Jupiter's position and motion
within time-velocity space. These geometric procedures anticipated the Oxford Calculators
, including the mean speed theorem
, by 14 centuries. South of Egypt the ancient Nubians
established a system of geometry including early versions of sun clocks.
In the 7th century BC, the Greek
mathematician Thales of Miletus
used geometry to solve problems such as calculating the height of pyramids and the distance of ships from the shore. He is credited with the first use of deductive reasoning applied to geometry, by deriving four corollaries to Thales' theorem
Pythagoras established the Pythagorean School
, which is credited with the first proof of the Pythagorean theorem
, though the statement of the theorem has a long history. Eudoxus
(408–c. 355 BC) developed the method of exhaustion
, which allowed the calculation of areas and volumes of curvilinear figures, as well as a theory of ratios that avoided the problem of incommensurable magnitudes
, which enabled subsequent geometers to make significant advances. Around 300 BC, geometry was revolutionized by Euclid, whose ''Elements
'', widely considered the most successful and influential textbook of all time, introduced mathematical rigor
through the axiomatic method
and is the earliest example of the format still used in mathematics today, that of definition, axiom, theorem, and proof. Although most of the contents of the ''Elements'' were already known, Euclid arranged them into a single, coherent logical framework.
The ''Elements'' was known to all educated people in the West until the middle of the 20th century and its contents are still taught in geometry classes today. Archimedes
(c. 287–212 BC) of Syracuse
used the method of exhaustion
to calculate the area
under the arc of a parabola
with the summation of an infinite series
, and gave remarkably accurate approximations of Pi
. He also studied the spiral
bearing his name and obtained formulas for the volume
s of surfaces of revolution
mathematicians also made many important contributions in geometry. The ''Satapatha Brahmana
'' (3rd century BC) contains rules for ritual geometric constructions that are similar to the ''Sulba Sutras
According to , the ''Śulba Sūtras'' contain "the earliest extant verbal expression of the Pythagorean Theorem in the world, although it had already been known to the Old Babylonians. They contain lists of Pythagorean triples
, which are particular cases of Diophantine equations
[: "The arithmetic content of the ''Śulva Sūtras'' consists of rules for finding Pythagorean triples such as (3, 4, 5), (5, 12, 13), (8, 15, 17), and (12, 35, 37). It is not certain what practical use these arithmetic rules had. The best conjecture is that they were part of religious ritual. A Hindu home was required to have three fires burning at three different altars. The three altars were to be of different shapes, but all three were to have the same area. These conditions led to certain "Diophantine" problems, a particular case of which is the generation of Pythagorean triples, so as to make one square integer equal to the sum of two others."]
In the Bakhshali manuscript
, there is a handful of geometric problems (including problems about volumes of irregular solids). The Bakhshali manuscript also "employs a decimal place value system with a dot for zero."
'' (499) includes the computation of areas and volumes.
wrote his astronomical work ''
'' in 628. Chapter 12, containing 66 Sanskrit
verses, was divided into two sections: "basic operations" (including cube roots, fractions, ratio and proportion, and barter) and "practical mathematics" (including mixture, mathematical series, plane figures, stacking bricks, sawing of timber, and piling of grain).
In the latter section, he stated his famous theorem on the diagonals of a cyclic quadrilateral
. Chapter 12 also included a formula for the area of a cyclic quadrilateral (a generalization of Heron's formula
), as well as a complete description of rational triangle
s (''i.e.'' triangles with rational sides and rational areas).
In the Middle Ages
, mathematics in medieval Islam
contributed to the development of geometry, especially algebraic geometry
(b. 853) conceived the idea of reducing geometrical problems such as duplicating the cube to problems in algebra. Thābit ibn Qurra
(known as Thebit in Latin
) (836–901) dealt with arithmetic
operations applied to ratio
s of geometrical quantities, and contributed to the development of analytic geometry
(1048–1131) found geometric solutions to cubic equation
s. The theorems of Ibn al-Haytham
(Alhazen), Omar Khayyam and Nasir al-Din al-Tusi
s, including the Lambert quadrilateral
and Saccheri quadrilateral
, were early results in hyperbolic geometry
, and along with their alternative postulates, such as Playfair's axiom
, these works had a considerable influence on the development of non-Euclidean geometry among later European geometers, including Witelo
(c. 1230–c. 1314), Gersonides
, John Wallis
, and Giovanni Girolamo Saccheri
In the early 17th century, there were two important developments in geometry. The first was the creation of analytic geometry, or geometry with coordinates
s, by René Descartes
(1596–1650) and Pierre de Fermat
This was a necessary precursor to the development of calculus
and a precise quantitative science of physics
The second geometric development of this period was the systematic study of projective geometry
by Girard Desargues
Projective geometry studies properties of shapes which are unchanged under projections
, especially as they relate to artistic perspective
Two developments in geometry in the 19th century changed the way it had been studied previously.
These were the discovery of non-Euclidean geometries
by Nikolai Ivanovich Lobachevsky, János Bolyai and Carl Friedrich Gauss and of the formulation of symmetry
as the central consideration in the Erlangen Programme
of Felix Klein
(which generalized the Euclidean and non-Euclidean geometries). Two of the master geometers of the time were Bernhard Riemann
(1826–1866), working primarily with tools from mathematical analysis
, and introducing the Riemann surface
, and Henri Poincaré
, the founder of algebraic topology
and the geometric theory of dynamical system
s. As a consequence of these major changes in the conception of geometry, the concept of "space" became something rich and varied, and the natural background for theories as different as complex analysis
and classical mechanics
Important concepts in geometry
The following are some of the most important concepts in geometry.
took an abstract approach to geometry in his Elements
one of the most influential books ever written.
Euclid introduced certain axiom
s, or postulate
s, expressing primary or self-evident properties of points, lines, and planes.
He proceeded to rigorously deduce other properties by mathematical reasoning. The characteristic feature of Euclid's approach to geometry was its rigor, and it has come to be known as ''axiomatic'' or ''synthetic
At the start of the 19th century, the discovery of non-Euclidean geometries
by Nikolai Ivanovich Lobachevsky
(1792–1856), János Bolyai
(1802–1860), Carl Friedrich Gauss
(1777–1855) and others
led to a revival of interest in this discipline, and in the 20th century, David Hilbert
(1862–1943) employed axiomatic reasoning in an attempt to provide a modern foundation of geometry.
Points are considered fundamental objects in Euclidean geometry. They have been defined in a variety of ways, including Euclid's definition as 'that which has no part'
[''Euclid's Elements – All thirteen books in one volume'', Based on Heath's translation, Green Lion Press .]
and through the use of algebra or nested sets. In many areas of geometry, such as analytic geometry, differential geometry, and topology, all objects are considered to be built up from points. However, there has been some study of geometry without reference to points.
described a line as "breadthless length" which "lies equally with respect to the points on itself".
In modern mathematics, given the multitude of geometries, the concept of a line is closely tied to the way the geometry is described. For instance, in analytic geometry
, a line in the plane is often defined as the set of points whose coordinates satisfy a given linear equation
, but in a more abstract setting, such as incidence geometry
, a line may be an independent object, distinct from the set of points which lie on it. In differential geometry, a geodesic
is a generalization of the notion of a line to curved spaces
is a flat, two-dimensional surface that extends infinitely far.
Planes are used in every area of geometry. For instance, planes can be studied as a topological surface
without reference to distances or angles;
[Munkres, James R. Topology. Vol. 2. Upper Saddle River: Prentice Hall, 2000.]
it can be studied as an affine space
, where collinearity and ratios can be studied but not distances; it can be studied as the complex plane
using techniques of complex analysis
; and so on.
defines a plane angle
as the inclination to each other, in a plane, of two lines which meet each other, and do not lie straight with respect to each other.
In modern terms, an angle is the figure formed by two rays
, called the ''sides'' of the angle, sharing a common endpoint, called the ''vertex
'' of the angle.
In Euclidean geometry
, angles are used to study polygon
s and triangle
s, as well as forming an object of study in their own right.
The study of the angles of a triangle or of angles in a unit circle
forms the basis of trigonometry
In differential geometry
, the angles between plane curve
s or space curve
s or surfaces
can be calculated using the derivative
[Stewart, James (2012). ''Calculus: Early Transcendentals'', 7th ed., Brooks Cole Cengage Learning. ]
is a 1-dimensional object that may be straight (like a line) or not; curves in 2-dimensional space are called plane curve
s and those in 3-dimensional space are called space curve
In topology, a curve is defined by a function from an interval of the real numbers to another space.
In differential geometry, the same definition is used, but the defining function is required to be differentiable
Algebraic geometry studies algebraic curve
s, which are defined as algebraic varieties
is a two-dimensional object, such as a sphere or paraboloid. In differential geometry
[Do Carmo, Manfredo Perdigao, and Manfredo Perdigao Do Carmo. Differential geometry of curves and surfaces. Vol. 2. Englewood Cliffs: Prentice-hall, 1976.]
surfaces are described by two-dimensional 'patches' (or neighborhoods
) that are assembled by diffeomorphism
s or homeomorphism
s, respectively. In algebraic geometry, surfaces are described by polynomial equation
is a generalization of the concepts of curve and surface. In topology
, a manifold is a topological space
where every point has a neighborhood
that is homeomorphic
to Euclidean space.
In differential geometry
, a differentiable manifold
is a space where each neighborhood is diffeomorphic
to Euclidean space.
Manifolds are used extensively in physics, including in general relativity and string theory.
Length, area, and volume
Length, area, and volume describe the size or extent of an object in one dimension, two dimension, and three dimensions respectively.
In Euclidean geometry and analytic geometry, the length of a line segment can often be calculated by the Pythagorean theorem.
Area and volume can be defined as fundamental quantities separate from length, or they can be described and calculated in terms of lengths in a plane or 3-dimensional space. Mathematicians have found many explicit formulas for area and formulas for volume of various geometric objects. In calculus, area and volume can be defined in terms of integrals, such as the Riemann integral or the Lebesgue integral.
Metrics and measures
The concept of length or distance can be generalized, leading to the idea of metrics. For instance, the Euclidean metric measures the distance between points in the Euclidean plane, while the hyperbolic metric measures the distance in the hyperbolic plane. Other important examples of metrics include the Lorentz metric of special relativity and the semi-Riemannian metrics of general relativity.
In a different direction, the concepts of length, area and volume are extended by measure theory, which studies methods of assigning a size or ''measure'' to sets, where the measures follow rules similar to those of classical area and volume.
Congruence and similarity
Congruence and similarity are concepts that describe when two shapes have similar characteristics.
In Euclidean geometry, similarity is used to describe objects that have the same shape, while congruence is used to describe objects that are the same in both size and shape. Hilbert, in his work on creating a more rigorous foundation for geometry, treated congruence as an undefined term whose properties are defined by axioms.
Congruence and similarity are generalized in transformation geometry, which studies the properties of geometric objects that are preserved by different kinds of transformations.
Compass and straightedge constructions
Classical geometers paid special attention to constructing geometric objects that had been described in some other way. Classically, the only instruments allowed in geometric constructions are the compass and straightedge. Also, every construction had to be complete in a finite number of steps. However, some problems turned out to be difficult or impossible to solve by these means alone, and ingenious constructions using parabolas and other curves, as well as mechanical devices, were found.
Where the traditional geometry allowed dimensions 1 (a line), 2 (a plane) and 3 (our ambient world conceived of as three-dimensional space), mathematicians and physicists have used higher dimensions for nearly two centuries.
One example of a mathematical use for higher dimensions is the configuration space of a physical system, which has a dimension equal to the system's degrees of freedom. For instance, the configuration of a screw can be described by five coordinates.
In general topology, the concept of dimension has been extended from natural numbers, to infinite dimension (Hilbert spaces, for example) and positive real numbers (in fractal geometry). In algebraic geometry, the dimension of an algebraic variety has received a number of apparently different definitions, which are all equivalent in the most common cases.
The theme of symmetry in geometry is nearly as old as the science of geometry itself.
Symmetric shapes such as the circle, regular polygons and platonic solids held deep significance for many ancient philosophers and were investigated in detail before the time of Euclid. Symmetric patterns occur in nature and were artistically rendered in a multitude of forms, including the graphics of Leonardo da Vinci, M. C. Escher, and others. In the second half of the 19th century, the relationship between symmetry and geometry came under intense scrutiny. Felix Klein's Erlangen program proclaimed that, in a very precise sense, symmetry, expressed via the notion of a transformation group, determines what geometry ''is''. Symmetry in classical Euclidean geometry is represented by congruences and rigid motions, whereas in projective geometry an analogous role is played by collineations, geometric transformations that take straight lines into straight lines. However it was in the new geometries of Bolyai and Lobachevsky, Riemann, Clifford and Klein, and Sophus Lie that Klein's idea to 'define a geometry via its symmetry group' found its inspiration. Both discrete and continuous symmetries play prominent roles in geometry, the former in topology and geometric group theory, the latter in Lie theory and Riemannian geometry.
A different type of symmetry is the principle of duality in projective geometry, among other fields. This meta-phenomenon can roughly be described as follows: in any theorem, exchange ''point'' with ''plane'', ''join'' with ''meet'', ''lies in'' with ''contains'', and the result is an equally true theorem. A similar and closely related form of duality exists between a vector space and its dual space.
Euclidean geometry is geometry in its classical sense.
As it models the space of the physical world, it is used in many scientific areas, such as mechanics, astronomy, crystallography, and many technical fields, such as engineering, architecture, geodesy, aerodynamics, and navigation. The mandatory educational curriculum of the majority of nations includes the study of Euclidean concepts such as points, lines, planes, angles, triangles, congruence, similarity, solid figures, circles, and analytic geometry. [Schmidt, W., Houang, R., & Cogan, L. (2002). "A coherent curriculum". ''American Educator'', 26(2), 1–18.]
Differential geometry uses techniques of calculus and linear algebra to study problems in geometry.
It has applications in physics, econometrics, and bioinformatics, among others.
In particular, differential geometry is of importance to mathematical physics due to Albert Einstein's general relativity postulation that the universe is curved. Differential geometry can either be ''intrinsic'' (meaning that the spaces it considers are smooth manifolds whose geometric structure is governed by a Riemannian metric, which determines how distances are measured near each point) or ''extrinsic'' (where the object under study is a part of some ambient flat Euclidean space).
Euclidean geometry was not the only historical form of geometry studied. Spherical geometry has long been used by astronomers, astrologers, and navigators.
Immanuel Kant argued that there is only one, ''absolute'', geometry, which is known to be true ''a priori'' by an inner faculty of mind: Euclidean geometry was synthetic a priori. This view was at first somewhat challenged by thinkers such as Saccheri, then finally overturned by the revolutionary discovery of non-Euclidean geometry in the works of Bolyai, Lobachevsky, and Gauss (who never published his theory). They demonstrated that ordinary Euclidean space is only one possibility for development of geometry. A broad vision of the subject of geometry was then expressed by Riemann in his 1867 inauguration lecture ''Über die Hypothesen, welche der Geometrie zu Grunde liegen'' (''On the hypotheses on which geometry is based''), published only after his death. Riemann's new idea of space proved crucial in Albert Einstein's general relativity theory. Riemannian geometry, which considers very general spaces in which the notion of length is defined, is a mainstay of modern geometry.
Topology is the field concerned with the properties of continuous mappings,
and can be considered a generalization of Euclidean geometry. In practice, topology often means dealing with large-scale properties of spaces, such as connectedness and compactness.
The field of topology, which saw massive development in the 20th century, is in a technical sense a type of transformation geometry, in which transformations are homeomorphisms. This has often been expressed in the form of the saying 'topology is rubber-sheet geometry'. Subfields of topology include geometric topology, differential topology, algebraic topology and general topology.
The field of algebraic geometry developed from the Cartesian geometry of co-ordinates.
It underwent periodic periods of growth, accompanied by the creation and study of projective geometry, birational geometry, algebraic varieties, and commutative algebra, among other topics. From the late 1950s through the mid-1970s it had undergone major foundational development, largely due to work of Jean-Pierre Serre and Alexander Grothendieck. This led to the introduction of schemes and greater emphasis on topological methods, including various cohomology theories. One of seven Millennium Prize problems, the Hodge conjecture, is a question in algebraic geometry. Wiles' proof of Fermat's Last Theorem uses advanced methods of algebraic geometry for solving a long-standing problem of number theory.
In general, algebraic geometry studies geometry through the use of concepts in commutative algebra such as multivariate polynomials. It has applications in many areas, including cryptography and string theory.
Complex geometry studies the nature of geometric structures modelled on, or arising out of, the complex plane. Complex geometry lies at the intersection of differential geometry, algebraic geometry, and analysis of several complex variables, and has found applications to string theory and mirror symmetry.
Complex geometry first appeared as a distinct area of study in the work of Bernhard Riemann in his study of Riemann surfaces. Work in the spirit of Riemann was carried out by the Italian school of algebraic geometry in the early 1900s. Contemporary treatment of complex geometry began with the work of Jean-Pierre Serre, who introduced the concept of sheaves to the subject, and illuminated the relations between complex geometry and algebraic geometry.
The primary objects of study in complex geometry are complex manifolds, complex algebraic varieties, and complex analytic varieties, and holomorphic vector bundles and coherent sheaves over these spaces. Special examples of spaces studied in complex geometry include Riemann surfaces, and Calabi-Yau manifolds, and these spaces find uses in string theory. In particular, worldsheets of strings are modelled by Riemann surfaces, and superstring theory predicts that the extra 6 dimensions of 10 dimensional spacetime may be modelled by Calabi-Yau manifolds.
Discrete geometry is a subject that has close connections with convex geometry.
It is concerned mainly with questions of relative position of simple geometric objects, such as points, lines and circles. Examples include the study of sphere packings, triangulations, the Kneser-Poulsen conjecture, etc. It shares many methods and principles with combinatorics.
Computational geometry deals with algorithms and their implementations for manipulating geometrical objects. Important problems historically have included the travelling salesman problem, minimum spanning trees, hidden-line removal, and linear programming.
Although being a young area of geometry, it has many applications in computer vision, image processing, computer-aided design, medical imaging, etc.
Geometric group theory
The Cayley graph of the free group on two generators ''a'' and ''b''
Geometric group theory uses large-scale geometric techniques to study finitely generated groups.
It is closely connected to low-dimensional topology, such as in Grigori Perelman's proof of the Geometrization conjecture, which included the proof of the Poincaré conjecture, a Millennium Prize Problem.
Geometric group theory often revolves around the Cayley graph, which is a geometric representation of a group. Other important topics include quasi-isometries, Gromov-hyperbolic groups, and right angled Artin groups.
Convex geometry investigates convex shapes in the Euclidean space and its more abstract analogues, often using techniques of real analysis and discrete mathematics.
It has close connections to convex analysis, optimization and functional analysis and important applications in number theory.
Convex geometry dates back to antiquity. Archimedes gave the first known precise definition of convexity. The isoperimetric problem, a recurring concept in convex geometry, was studied by the Greeks as well, including Zenodorus. Archimedes, Plato, Euclid, and later Kepler and Coxeter all studied convex polytopes and their properties. From the 19th century on, mathematicians have studied other areas of convex mathematics, including higher-dimensional polytopes, volume and surface area of convex bodies, Gaussian curvature, algorithms, tilings and lattices.
Geometry has found applications in many fields, some of which are described below.
Mathematics and art are related in a variety of ways. For instance, the theory of perspective showed that there is more to geometry than just the metric properties of figures: perspective is the origin of projective geometry.
Artists have long used concepts of proportion in design. Vitruvius developed a complicated theory of ''ideal proportions'' for the human figure. These concepts have been used and adapted by artists from Michelangelo to modern comic book artists.
The golden ratio is a particular proportion that has had a controversial role in art. Often claimed to be the most aesthetically pleasing ratio of lengths, it is frequently stated to be incorporated into famous works of art, though the most reliable and unambiguous examples were made deliberately by artists aware of this legend.
Tilings, or tessellations, have been used in art throughout history. Islamic art makes frequent use of tessellations, as did the art of M. C. Escher. Escher's work also made use of hyperbolic geometry.
Cézanne advanced the theory that all images can be built up from the sphere, the cone, and the cylinder. This is still used in art theory today, although the exact list of shapes varies from author to author.
Geometry has many applications in architecture. In fact, it has been said that geometry lies at the core of architectural design.
Applications of geometry to architecture include the use of projective geometry to create forced perspective, the use of conic sections in constructing domes and similar objects, the use of tessellations, and the use of symmetry.
The field of astronomy, especially as it relates to mapping the positions of stars and planets on the celestial sphere and describing the relationship between movements of celestial bodies, have served as an important source of geometric problems throughout history.
Riemannian geometry and pseudo-Riemannian geometry are used in general relativity. String theory makes use of several variants of geometry, as does quantum information theory.
Other fields of mathematics
Calculus was strongly influenced by geometry.
For instance, the introduction of coordinates by René Descartes and the concurrent developments of algebra marked a new stage for geometry, since geometric figures such as plane curves could now be represented analytically in the form of functions and equations. This played a key role in the emergence of infinitesimal calculus in the 17th century. Analytic geometry continues to be a mainstay of pre-calculus and calculus curriculum.
Another important area of application is number theory. In ancient Greece the Pythagoreans considered the role of numbers in geometry. However, the discovery of incommensurable lengths contradicted their philosophical views. Since the 19th century, geometry has been used for solving problems in number theory, for example through the geometry of numbers or, more recently, scheme theory, which is used in Wiles's proof of Fermat's Last Theorem.
* List of geometers
** :Category:Algebraic geometers
** :Category:Differential geometers
* List of formulas in elementary geometry
* List of geometry topics
* List of important publications in geometry
* Lists of mathematics topics
* Descriptive geometry
* Finite geometry
* ''Flatland'', a book written by Edwin Abbott Abbott about two- and three-dimensional space, to understand the concept of four dimensions
* List of interactive geometry software
* Molecular geometry
* A geometry course from Wikiversity
''Unusual Geometry Problems''
''The Math Forum'' – Geometry
Nature Precedings – ''Pegs and Ropes Geometry at Stonehenge''
* ttps://web.archive.org/web/20071004174210/http://www.gresham.ac.uk/event.asp?PageId=45&EventId=618 "4000 Years of Geometry" lecture by Robin Wilson given at Gresham College, 3 October 2007 (available for MP3 and MP4 download as well as a text file)
Finitism in Geometry
at the Stanford Encyclopedia of Philosophy
Interactive geometry reference with hundreds of applets
at Khan Academy